Why won't the marker center?

Are you trying to change your scatter plot squares to round dots?

If so, select you plot, ungroup it until all of the plot points are ungrouped. Next, select all of your plot points, then go to the Effect menu, select "convert to shape", then choose ellipse. You can enter the size you need of the ellipse, then click ok. With the ellipses still selected you can assign stroke and fill color.

It's pretty obvious that you have a bounding box around the circle.

The only way I can get it to offset in the preview is by having uneven boxes in around the circle.

However, this doesn't seem to be what you have here.

It's a bit of a stumper - be nice to see the files if you get a chance to have them downloadable.

Out of curiosity - have you tried resetting your preferences?

To restore preferences quickly using a keyboard shortcut. Press and hold Alt+Control+Shift (Windows) or Option+Command+Shift (macOS) as you start Illustrator. You are prompted to delete the current settings. The new preferences files are created the next time you start Illustrator.

The markers disappear in the graph when I select the design, but I am not sure if it is related or not.

Is it just the preview?

Or does this happen in the graph as well?